3 problems

Integral Borel basis conjecture. The elements $E_AH_B$ with $A\in{\mathbb N}^{\Phi^+}$, $B\in{\mathbb N}_{i_0}^n$, and $|A|+|B|\le d$ form an $\mathcal A$-basis of…

Quantum Cartan-generator generation conjecture. The elements $\begin{bmatrix}K_i\\m\end{bmatrix}$ lie in the subring generated by the $E_i^{(m)}$ and $F_i^{(m)}$; consequently, one…

Let $R$ be the integral coefficient ring, let $\overline R$ be the corresponding coefficient ring for the construction with $e$ replaced by $e+1$, and let $O^{\nu,\Delta}_{\mu,R}$…